Cost and convenience are the primary drivers here. This is a group of relatively inexperienced drivers, making a very very long journey with a small baby.
I'm nervous of them driving up. I think it would be hell.
What dates are you (well, they) looking at, and how much flexibility around those dates will they have?
As others have said, if you're booking too far out no advance (ie discounted) tickets are available, only open (ie flexible but full price) ones. The LNER site only has ticket availability until April 23 - timetables are fixed until then so advance tickets are available - but other sites may be offering (only) flexible bookings after that.
Right now, I could book tickets for two adults and a child to travel tomorrow and return next Thursday for £252 (which a £30 family railcard would reduce to £190).
The same prices are available for travel out on 15/04 and back on 22/04 on fixed tickets, but flexible fares look like £350 each way for Anytime Singles or £365 for an off-peak return (£211.40 each way/£220.40 return with a railcard). (The limited availability and lack of variation on Advance ticket prices makes me think there won't be much to be gained from flexibility in their travel dates - I guess it'll be the same price whenever they go.)
Those prices are for the party, not per person. The baby doesn't need a ticket; their son needs a child ticket if he's 5 or over.
A Family and Friends railcard costs £30 for a year, and gives 1-4 adults and 1-4 children travelling together 1/3 off the adult fares and 60% off the child fares (kids used to be a £1 flat fare, but that was a long time ago). It'll be worth it for this one journey: NB that the adults must be travelling with a fare-paying child, so it will almost certainly be worth buying a ticket for their son even if he's 3 or 4 and could otherwise travel free.
ETA: I've just re-read your post, and it's your son not theirs, so three adults and a baby rather than two adults, a baby and a child. You also mention the Family railcard but quote undiscounted prices, so I suspect you haven't included the child ticket which, while not needed by the baby, is necessary to trigger the discount.
Make the prices without a railcard (3 adults, one unticketed baby) £302.40 (Advance); £420 each way (Anytime Single); £438 (Offpeak return); £547.50 (Anytime return).
Alternatively, with a railcard (3 adults, one child ticket), it's £273.20 (Advance); £303.80 each way (Anytime Single); £316.75 (Offpeak return); £396 (Anytime return). That means that on Advance tickets, it would cost them an extra 80p to get a railcard valid for the next 12 months.